Free actions of some compact groups on Milnor manifolds

TL;DR

This paper studies free actions of certain compact groups on Milnor manifolds, computes their orbit space cohomology, and derives applications like Borsuk-Ulam type results and bounds on the Schwarz genus.

Contribution

It provides the first computation of mod 2 cohomology of orbit spaces under free group actions on Milnor manifolds and applies these results to topological coincidence problems.

Findings

Computed mod 2 cohomology algebra of orbit spaces

Established Borsuk-Ulam type theorems for these actions

Derived lower bounds on Schwarz genus for complex Milnor manifolds

Abstract

In this paper, we investigate free actions of some compact groups on cohomology real and complex Milnor manifolds. More precisely, we compute the mod 2 cohomology algebra of the orbit space of an arbitrary free $\mathbb{Z}_2$ and $\mathbb{S}^1$-action on a compact Hausdorff space with mod 2 cohomology algebra of a real or a complex Milnor manifold. As applications, we deduce some Borsuk-Ulam type results for equivariant maps between spheres and these spaces. For the complex case, we obtain a lower bound on the Schwarz genus, which further establishes the existence of coincidence points for maps to the Euclidean plane.